1,1,58,0,1.405308," ","integrate(tan(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{15 \, x \cos\left(x\right)^{3} + 23 \, \cos\left(x\right)^{4} - 19 \, \cos\left(x\right)^{2} + {\left(15 \, x \cos\left(x\right)^{3} - 8 \, \cos\left(x\right)^{2} + 1\right)} \sin\left(x\right) + 4}{15 \, {\left(a \cos\left(x\right)^{3} \sin\left(x\right) + a \cos\left(x\right)^{3}\right)}}"," ",0,"1/15*(15*x*cos(x)^3 + 23*cos(x)^4 - 19*cos(x)^2 + (15*x*cos(x)^3 - 8*cos(x)^2 + 1)*sin(x) + 4)/(a*cos(x)^3*sin(x) + a*cos(x)^3)","A",0
2,1,71,0,0.519359," ","integrate(tan(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{10 \, \cos\left(x\right)^{2} + 11 \, {\left(\cos\left(x\right)^{2} \sin\left(x\right) + \cos\left(x\right)^{2}\right)} \log\left(\sin\left(x\right) + 1\right) + 5 \, {\left(\cos\left(x\right)^{2} \sin\left(x\right) + \cos\left(x\right)^{2}\right)} \log\left(-\sin\left(x\right) + 1\right) + 6 \, \sin\left(x\right) + 2}{16 \, {\left(a \cos\left(x\right)^{2} \sin\left(x\right) + a \cos\left(x\right)^{2}\right)}}"," ",0,"1/16*(10*cos(x)^2 + 11*(cos(x)^2*sin(x) + cos(x)^2)*log(sin(x) + 1) + 5*(cos(x)^2*sin(x) + cos(x)^2)*log(-sin(x) + 1) + 6*sin(x) + 2)/(a*cos(x)^2*sin(x) + a*cos(x)^2)","A",0
3,1,38,0,0.728284," ","integrate(tan(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{3 \, x \cos\left(x\right) + 4 \, \cos\left(x\right)^{2} + {\left(3 \, x \cos\left(x\right) - 1\right)} \sin\left(x\right) - 2}{3 \, {\left(a \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)}}"," ",0,"-1/3*(3*x*cos(x) + 4*cos(x)^2 + (3*x*cos(x) - 1)*sin(x) - 2)/(a*cos(x)*sin(x) + a*cos(x))","A",0
4,1,35,0,0.533700," ","integrate(tan(x)/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{3 \, {\left(\sin\left(x\right) + 1\right)} \log\left(\sin\left(x\right) + 1\right) + {\left(\sin\left(x\right) + 1\right)} \log\left(-\sin\left(x\right) + 1\right) + 2}{4 \, {\left(a \sin\left(x\right) + a\right)}}"," ",0,"-1/4*(3*(sin(x) + 1)*log(sin(x) + 1) + (sin(x) + 1)*log(-sin(x) + 1) + 2)/(a*sin(x) + a)","A",0
5,1,9,0,0.700227," ","integrate(cot(x)/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{a}"," ",0,"log(sin(x) + 1)/a","A",0
6,1,25,0,0.490433," ","integrate(cot(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{2 \, x + \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, a}"," ",0,"-1/2*(2*x + log(1/2*cos(x) + 1/2) - log(-1/2*cos(x) + 1/2))/a","A",0
7,1,19,0,0.603979," ","integrate(cot(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{\log\left(\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right) + 1}{a \sin\left(x\right)}"," ",0,"-(log(1/2*sin(x))*sin(x) + 1)/(a*sin(x))","A",0
8,1,64,0,0.480818," ","integrate(cot(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{4 \, x \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(\cos\left(x\right)^{2} - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 4 \, \cos\left(x\right) \sin\left(x\right) - 4 \, x + 2 \, \cos\left(x\right)}{4 \, {\left(a \cos\left(x\right)^{2} - a\right)}}"," ",0,"1/4*(4*x*cos(x)^2 + (cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) - (cos(x)^2 - 1)*log(-1/2*cos(x) + 1/2) - 4*cos(x)*sin(x) - 4*x + 2*cos(x))/(a*cos(x)^2 - a)","B",0
9,1,45,0,0.722084," ","integrate(cot(x)^5/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{6 \, {\left(\cos\left(x\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right) + 6 \, \cos\left(x\right)^{2} - 3 \, \sin\left(x\right) - 4}{6 \, {\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}"," ",0,"1/6*(6*(cos(x)^2 - 1)*log(1/2*sin(x))*sin(x) + 6*cos(x)^2 - 3*sin(x) - 4)/((a*cos(x)^2 - a)*sin(x))","A",0
10,1,104,0,0.581219," ","integrate(cot(x)^6/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{48 \, x \cos\left(x\right)^{4} - 96 \, x \cos\left(x\right)^{2} + 30 \, \cos\left(x\right)^{3} + 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} + 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 16 \, {\left(4 \, \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)\right)} \sin\left(x\right) + 48 \, x - 18 \, \cos\left(x\right)}{48 \, {\left(a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a\right)}}"," ",0,"-1/48*(48*x*cos(x)^4 - 96*x*cos(x)^2 + 30*cos(x)^3 + 9*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) - 9*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/2*cos(x) + 1/2) - 16*(4*cos(x)^3 - 3*cos(x))*sin(x) + 48*x - 18*cos(x))/(a*cos(x)^4 - 2*a*cos(x)^2 + a)","B",0
11,1,70,0,0.632027," ","integrate(cot(x)^7/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{60 \, \cos\left(x\right)^{4} + 60 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} + 1\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right) - 80 \, \cos\left(x\right)^{2} - 15 \, {\left(4 \, \cos\left(x\right)^{2} - 3\right)} \sin\left(x\right) + 32}{60 \, {\left(a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a\right)} \sin\left(x\right)}"," ",0,"-1/60*(60*cos(x)^4 + 60*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*sin(x))*sin(x) - 80*cos(x)^2 - 15*(4*cos(x)^2 - 3)*sin(x) + 32)/((a*cos(x)^4 - 2*a*cos(x)^2 + a)*sin(x))","A",0
12,1,260,0,1.422868," ","integrate(tan(x)^5/(a+b*csc(x)),x, algorithm=""fricas"")","\frac{16 \, b^{6} \cos\left(x\right)^{4} \log\left(a \sin\left(x\right) + b\right) + 4 \, a^{6} - 8 \, a^{4} b^{2} + 4 \, a^{2} b^{4} - {\left(8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(x\right)^{4} \log\left(\sin\left(x\right) + 1\right) - {\left(8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right)} \cos\left(x\right)^{4} \log\left(-\sin\left(x\right) + 1\right) - 8 \, {\left(2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(x\right)^{2} - 2 \, {\left(2 \, a^{5} b - 4 \, a^{3} b^{3} + 2 \, a b^{5} - {\left(5 \, a^{5} b - 14 \, a^{3} b^{3} + 9 \, a b^{5}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{16 \, {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(x\right)^{4}}"," ",0,"1/16*(16*b^6*cos(x)^4*log(a*sin(x) + b) + 4*a^6 - 8*a^4*b^2 + 4*a^2*b^4 - (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(x)^4*log(sin(x) + 1) - (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cos(x)^4*log(-sin(x) + 1) - 8*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4)*cos(x)^2 - 2*(2*a^5*b - 4*a^3*b^3 + 2*a*b^5 - (5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cos(x)^2)*sin(x))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(x)^4)","A",0
13,1,144,0,1.868122," ","integrate(tan(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","\frac{4 \, b^{4} \cos\left(x\right)^{2} \log\left(a \sin\left(x\right) + b\right) + 2 \, a^{4} - 2 \, a^{2} b^{2} + {\left(2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) - 2 \, {\left(a^{3} b - a b^{3}\right)} \sin\left(x\right)}{4 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(x\right)^{2}}"," ",0,"1/4*(4*b^4*cos(x)^2*log(a*sin(x) + b) + 2*a^4 - 2*a^2*b^2 + (2*a^4 + a^3*b - 4*a^2*b^2 - 3*a*b^3)*cos(x)^2*log(sin(x) + 1) + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*cos(x)^2*log(-sin(x) + 1) - 2*(a^3*b - a*b^3)*sin(x))/((a^5 - 2*a^3*b^2 + a*b^4)*cos(x)^2)","A",0
14,1,58,0,1.249022," ","integrate(tan(x)/(a+b*csc(x)),x, algorithm=""fricas"")","\frac{2 \, b^{2} \log\left(a \sin\left(x\right) + b\right) - {\left(a^{2} + a b\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(a^{2} - a b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{3} - a b^{2}\right)}}"," ",0,"1/2*(2*b^2*log(a*sin(x) + b) - (a^2 + a*b)*log(sin(x) + 1) - (a^2 - a*b)*log(-sin(x) + 1))/(a^3 - a*b^2)","A",0
15,1,11,0,0.549327," ","integrate(cot(x)/(a+b*csc(x)),x, algorithm=""fricas"")","\frac{\log\left(a \sin\left(x\right) + b\right)}{a}"," ",0,"log(a*sin(x) + b)/a","A",0
16,1,47,0,0.476945," ","integrate(cot(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{a^{2} \log\left(-\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \log\left(a \sin\left(x\right) + b\right) \sin\left(x\right) + a b}{a b^{2} \sin\left(x\right)}"," ",0,"-(a^2*log(-1/2*sin(x))*sin(x) - (a^2 - b^2)*log(a*sin(x) + b)*sin(x) + a*b)/(a*b^2*sin(x))","A",0
17,1,151,0,0.511060," ","integrate(cot(x)^5/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{3 \, a^{2} b^{2} \sin\left(x\right) - 6 \, a^{3} b + 10 \, a b^{3} + 6 \, {\left(a^{3} b - 2 \, a b^{3}\right)} \cos\left(x\right)^{2} + 6 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(a \sin\left(x\right) + b\right) \sin\left(x\right) - 6 \, {\left(a^{4} - 2 \, a^{2} b^{2} - {\left(a^{4} - 2 \, a^{2} b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right)}{6 \, {\left(a b^{4} \cos\left(x\right)^{2} - a b^{4}\right)} \sin\left(x\right)}"," ",0,"-1/6*(3*a^2*b^2*sin(x) - 6*a^3*b + 10*a*b^3 + 6*(a^3*b - 2*a*b^3)*cos(x)^2 + 6*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*log(a*sin(x) + b)*sin(x) - 6*(a^4 - 2*a^2*b^2 - (a^4 - 2*a^2*b^2)*cos(x)^2)*log(1/2*sin(x))*sin(x))/((a*b^4*cos(x)^2 - a*b^4)*sin(x))","B",0
18,1,327,0,0.559378," ","integrate(cot(x)^7/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{60 \, a^{5} b - 160 \, a^{3} b^{3} + 132 \, a b^{5} + 60 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(x\right)^{4} - 20 \, {\left(6 \, a^{5} b - 17 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(x\right)^{2} - 60 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6} + {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{4} - 2 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{2}\right)} \log\left(a \sin\left(x\right) + b\right) \sin\left(x\right) + 60 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(x\right)^{4} - 2 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \sin\left(x\right)\right) \sin\left(x\right) - 15 \, {\left(2 \, a^{4} b^{2} - 5 \, a^{2} b^{4} - 2 \, {\left(a^{4} b^{2} - 3 \, a^{2} b^{4}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{60 \, {\left(a b^{6} \cos\left(x\right)^{4} - 2 \, a b^{6} \cos\left(x\right)^{2} + a b^{6}\right)} \sin\left(x\right)}"," ",0,"-1/60*(60*a^5*b - 160*a^3*b^3 + 132*a*b^5 + 60*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cos(x)^4 - 20*(6*a^5*b - 17*a^3*b^3 + 15*a*b^5)*cos(x)^2 - 60*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^2)*log(a*sin(x) + b)*sin(x) + 60*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cos(x)^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cos(x)^2)*log(-1/2*sin(x))*sin(x) - 15*(2*a^4*b^2 - 5*a^2*b^4 - 2*(a^4*b^2 - 3*a^2*b^4)*cos(x)^2)*sin(x))/((a*b^6*cos(x)^4 - 2*a*b^6*cos(x)^2 + a*b^6)*sin(x))","B",0
19,1,483,0,0.576120," ","integrate(tan(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a^{2} - b^{2}} b^{5} \cos\left(x\right)^{3} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - 2 \, a^{5} b + 4 \, a^{3} b^{3} - 2 \, a b^{5} + 6 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} x \cos\left(x\right)^{3} + 6 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} - {\left(4 \, a^{6} - 11 \, a^{4} b^{2} + 7 \, a^{2} b^{4}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{6 \, {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(x\right)^{3}}, \frac{3 \, \sqrt{-a^{2} + b^{2}} b^{5} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) \cos\left(x\right)^{3} - a^{5} b + 2 \, a^{3} b^{3} - a b^{5} + 3 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} x \cos\left(x\right)^{3} + 3 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(x\right)^{2} + {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} - {\left(4 \, a^{6} - 11 \, a^{4} b^{2} + 7 \, a^{2} b^{4}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{3 \, {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(x\right)^{3}}\right]"," ",0,"[1/6*(3*sqrt(a^2 - b^2)*b^5*cos(x)^3*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - 2*a^5*b + 4*a^3*b^3 - 2*a*b^5 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cos(x)^3 + 6*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(x)^2 + 2*(a^6 - 2*a^4*b^2 + a^2*b^4 - (4*a^6 - 11*a^4*b^2 + 7*a^2*b^4)*cos(x)^2)*sin(x))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(x)^3), 1/3*(3*sqrt(-a^2 + b^2)*b^5*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*cos(x)^3 - a^5*b + 2*a^3*b^3 - a*b^5 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cos(x)^3 + 3*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(x)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4 - (4*a^6 - 11*a^4*b^2 + 7*a^2*b^4)*cos(x)^2)*sin(x))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(x)^3)]","A",0
20,1,296,0,0.471415," ","integrate(tan(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a^{2} - b^{2}} b^{3} \cos\left(x\right) \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + 2 \, a^{3} b - 2 \, a b^{3} + 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} x \cos\left(x\right) - 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(x\right)}, \frac{\sqrt{-a^{2} + b^{2}} b^{3} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) \cos\left(x\right) - a^{3} b + a b^{3} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} x \cos\left(x\right) + {\left(a^{4} - a^{2} b^{2}\right)} \sin\left(x\right)}{{\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(x\right)}\right]"," ",0,"[-1/2*(sqrt(a^2 - b^2)*b^3*cos(x)*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 2*a^3*b - 2*a*b^3 + 2*(a^4 - 2*a^2*b^2 + b^4)*x*cos(x) - 2*(a^4 - a^2*b^2)*sin(x))/((a^5 - 2*a^3*b^2 + a*b^4)*cos(x)), (sqrt(-a^2 + b^2)*b^3*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*cos(x) - a^3*b + a*b^3 - (a^4 - 2*a^2*b^2 + b^4)*x*cos(x) + (a^4 - a^2*b^2)*sin(x))/((a^5 - 2*a^3*b^2 + a*b^4)*cos(x))]","A",0
21,1,206,0,0.594821," ","integrate(cot(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[-\frac{2 \, b x + a \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - a \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right)}{2 \, a b}, -\frac{2 \, b x + a \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - a \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right)}{2 \, a b}\right]"," ",0,"[-1/2*(2*b*x + a*log(1/2*cos(x) + 1/2) - a*log(-1/2*cos(x) + 1/2) - sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)))/(a*b), -1/2*(2*b*x + a*log(1/2*cos(x) + 1/2) - a*log(-1/2*cos(x) + 1/2) - 2*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))))/(a*b)]","A",0
22,1,453,0,0.631892," ","integrate(cot(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{4 \, b^{3} x \cos\left(x\right)^{2} - 4 \, a^{2} b \cos\left(x\right) \sin\left(x\right) - 4 \, b^{3} x + 2 \, a b^{2} \cos\left(x\right) - 2 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a^{2} - b^{2}} \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + {\left(2 \, a^{3} - 3 \, a b^{2} - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(2 \, a^{3} - 3 \, a b^{2} - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a b^{3} \cos\left(x\right)^{2} - a b^{3}\right)}}, \frac{4 \, b^{3} x \cos\left(x\right)^{2} - 4 \, a^{2} b \cos\left(x\right) \sin\left(x\right) - 4 \, b^{3} x + 2 \, a b^{2} \cos\left(x\right) + 4 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) + {\left(2 \, a^{3} - 3 \, a b^{2} - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(2 \, a^{3} - 3 \, a b^{2} - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a b^{3} \cos\left(x\right)^{2} - a b^{3}\right)}}\right]"," ",0,"[1/4*(4*b^3*x*cos(x)^2 - 4*a^2*b*cos(x)*sin(x) - 4*b^3*x + 2*a*b^2*cos(x) - 2*((a^2 - b^2)*cos(x)^2 - a^2 + b^2)*sqrt(a^2 - b^2)*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + (2*a^3 - 3*a*b^2 - (2*a^3 - 3*a*b^2)*cos(x)^2)*log(1/2*cos(x) + 1/2) - (2*a^3 - 3*a*b^2 - (2*a^3 - 3*a*b^2)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a*b^3*cos(x)^2 - a*b^3), 1/4*(4*b^3*x*cos(x)^2 - 4*a^2*b*cos(x)*sin(x) - 4*b^3*x + 2*a*b^2*cos(x) + 4*((a^2 - b^2)*cos(x)^2 - a^2 + b^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + (2*a^3 - 3*a*b^2 - (2*a^3 - 3*a*b^2)*cos(x)^2)*log(1/2*cos(x) + 1/2) - (2*a^3 - 3*a*b^2 - (2*a^3 - 3*a*b^2)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a*b^3*cos(x)^2 - a*b^3)]","B",0
23,1,852,0,0.953800," ","integrate(cot(x)^6/(a+b*csc(x)),x, algorithm=""fricas"")","\left[-\frac{48 \, b^{5} x \cos\left(x\right)^{4} - 96 \, b^{5} x \cos\left(x\right)^{2} + 48 \, b^{5} x - 6 \, {\left(4 \, a^{3} b^{2} - 9 \, a b^{4}\right)} \cos\left(x\right)^{3} - 24 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2}\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + 6 \, {\left(4 \, a^{3} b^{2} - 7 \, a b^{4}\right)} \cos\left(x\right) + 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{4} - 2 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{4} - 2 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 16 \, {\left({\left(3 \, a^{4} b - 7 \, a^{2} b^{3}\right)} \cos\left(x\right)^{3} - 3 \, {\left(a^{4} b - 2 \, a^{2} b^{3}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{48 \, {\left(a b^{5} \cos\left(x\right)^{4} - 2 \, a b^{5} \cos\left(x\right)^{2} + a b^{5}\right)}}, -\frac{48 \, b^{5} x \cos\left(x\right)^{4} - 96 \, b^{5} x \cos\left(x\right)^{2} + 48 \, b^{5} x - 6 \, {\left(4 \, a^{3} b^{2} - 9 \, a b^{4}\right)} \cos\left(x\right)^{3} - 48 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) + 6 \, {\left(4 \, a^{3} b^{2} - 7 \, a b^{4}\right)} \cos\left(x\right) + 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{4} - 2 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{4} - 2 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 16 \, {\left({\left(3 \, a^{4} b - 7 \, a^{2} b^{3}\right)} \cos\left(x\right)^{3} - 3 \, {\left(a^{4} b - 2 \, a^{2} b^{3}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{48 \, {\left(a b^{5} \cos\left(x\right)^{4} - 2 \, a b^{5} \cos\left(x\right)^{2} + a b^{5}\right)}}\right]"," ",0,"[-1/48*(48*b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*a*b^4)*cos(x)^3 - 24*((a^4 - 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 6*(4*a^3*b^2 - 7*a*b^4)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2) + 16*((3*a^4*b - 7*a^2*b^3)*cos(x)^3 - 3*(a^4*b - 2*a^2*b^3)*cos(x))*sin(x))/(a*b^5*cos(x)^4 - 2*a*b^5*cos(x)^2 + a*b^5), -1/48*(48*b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*a*b^4)*cos(x)^3 - 48*((a^4 - 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + 6*(4*a^3*b^2 - 7*a*b^4)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2) + 16*((3*a^4*b - 7*a^2*b^3)*cos(x)^3 - 3*(a^4*b - 2*a^2*b^3)*cos(x))*sin(x))/(a*b^5*cos(x)^4 - 2*a*b^5*cos(x)^2 + a*b^5)]","B",0
